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Computational Methods for Modelling Stationary and Non-Stationary Interfaces at Multiple Scales

Mikhail Poluektov, University of Warwick

Lukasz Figiel, University of Warwick

Nikhil Chandra Admal, University of Illinois at Urbana-Champaign

Giacomo Po, University of Miami


There is a wide range of problems in continuum mechanics that involve both stationary and non-stationary interfaces. The latter problems include, for example, classical grain microstructure evolution, stress-induced solid-solid phase transitions, chemo-mechanics with localised chemical reaction fronts, mechanics of biomembranes and surfaces of growth. Physical behaviour of materials undergoing such microstructure evolution is an active area of research, as there is a significant industrial demand emerging from a wide range of engineering applications of phase-transforming materials in the areas such as microelectronics, battery components, sensors and actuators.

Kinetics of the non-stationary interfaces, such as grain/phase boundaries and transformation fronts, is governed by configurational forces, which strongly depend on mechanical stresses and deformations, which, in turn, depend on the interface position. Furthermore, phase-transforming materials often involve multi-physics processes. For example, twin boundaries in magnetic shape memory alloys (MSMAs) are affected both by mechanical stresses and external magnetic field. Another example can be chemical reaction fronts in oxides that are affected both by mechanical stresses and oxygen diffusion. Finally, these materials often undergo large deformations; therefore, the non-linear mechanical behaviour must also be considered.

From the mathematical point of view, there are two classical modelling paradigms that can be used for modelling materials with interfaces – the atomistic and the continuum approaches, which have their own advantages and disadvantages. The latter option is somewhat more challenging from the methodological point of view, especially in the case of non-stationary interfaces, where the considered problems are represented by systems of non-linear PDEs with discontinuities across time-dependent interfaces, velocity of which depend on the solution of the PDEs. During the past decades, a large variety of computational methods emerged in attempts to solve efficiently such problems. Most methods can be classified into two groups: the diffuse-interface methods, such as the phase-field method, and the sharp-interface methods. There are also some noteworthy modifications of the former, such as the so-called sharp phase-field method. The category of classical sharp-interface methods includes the boundary integral method, the standard FEM with finite-element mesh conforming to the interface and with remeshing each time the interface moves, and the fictitious-domain methods, where the mesh is static and the interfaces cut through the elements. Such methods include XFEM, GFEM, enhanced gradient FEM and CutFEM, popularity of which increased over the past years.

Aims of the mini-symposium

The aim of this mini-symposium is twofold. The first aim is to combine recent developments in computational methods for modelling kinetics of time-dependent interfaces, including surface-evolution algorithms. The topics will include both the diffuse-interface and the sharp-interface methods, with a specific emphasis devoted to the comparison of these frameworks and to the analysis of advantages and disadvantages of individual approaches. Furthermore, the mini-symposium will discuss various types of phase-transforming materials, especially materials involving multi-physics processes and materials evolving under extreme thermomechanical loads, as they require the solution of non-linear coupled problems. The second aim is to discuss multiscale aspects of interface modelling, more specifically, atomistic and continuum modelling approaches and transitions between them. This part will also discuss the coevolution of the bulk and the interfaces at multiple scales, for example, the interaction between grain boundaries and dislocations, bulk and surface diffusion in biomembranes, and graphene-metal interfaces.